**W**hile we enter physics to study the fascinating world of black holes, quarks and the quantum, the brutal truth is that mathematics is the central tool of the physicist. Gauss called mathematics the "Queen of the Sciences", and with good reason. If you don't have a solid grasp of mathematics, you aren't going to get very far.

One thing I noticed when getting my degrees in physics was that many of the students found math to be a painful "aside". In one case that really stands out in my memory, The student in question had thought he was interested in physics but didn't want to bother with the work of physics-which involves diving into the mathematics. But to become a good physicist-or a solid engineer-you need to bite the bullet and become a master of mathematics. It doesn't matter if you're going to be an astronomer, experimentalist, or engineer-in my view if you want to be the best at what you do in these fields, you should have a solid command of math. So if you are interested in physics but aren't a mathematical hot shot, how can you pull yourself to the top of the field? In my view, the answer is to view mathematics the way you would athletics. A friend of mine who shared this view coined the term "mental gymnastics" to characterize his outlook and study habits.

We all aren't Math Genuises

While for some students thinking mathematically comes natural, most of us aren't ready to master the intricacies of studying proofs when we're college freshmen. This article is written for those of us who aren't automatic math whiz kids. If you are a mere mortal who finds math a bit of work, don't be discouraged. It's my belief that average people can raise themselves up to become very good mathematicians with a little bit of hard work. What we need is some training--we need to train our minds to think mathematically. The best way to think about how you can get this done is to draw an analogy between math and athletics.

To master a sport you have to build your muscles and train your body to react in certain ways. For example, if you want to become a great basketball player, you could be lucky enough to be born Michael Jordan. But more likely, you'll have to work at building a basic skill set, and the truth is even players like Michael Jordan put extra work into their craft. Some activiities you might consider that could make you a better basketball player are

- Lifting weights to build muscle mass
- Run sprints to improve your ability to run up and down the court without getting tired
- Spend a large amount of time shooting free throws, doing layups and practicing basic skills like passing

It turns out that becoming a successful physicist or engineer is in many ways similar to athletics. OK, so suppose you want to study Hawking radiation and string theory, but you are not a hot shot mathematician and weren't the best student. Instead of just reading a bunch of books or lamenting the fact we aren't an

*Einsteinian*genius, what are the mathematical equivalents to lifting weights or running sprints we can do to improve our mathematical ability? In my view, we can begin by following two steps

*learn the basic rules first**repeat, repeat, repeat*

That is do tons of problems. In my view a student should start off simple. Don't try to understand the proofs. For example, in my recent book, "Calculus In Focus", I take the perspective that students need to learn math by following the formula: show, repeat, try it yourself. That is

*Show the student a given rule, like the product rule for derivatives**Focus on mastering calculational skills first. Do this by showing the student how to apply the rule with multiple examples.**Repeat, repeat, repeat. Do a given type of problem multiple times so that it becomes second nature.*

Once the "how" to solve problems is second nature, then go back for a deeper look at the material. Then learn the "why" and start learning the formality of mathematics through proofs and theorems. I use this approach to drill the central ideas of calculus in my book Calculus in Focus. More information can be found

In addition to the basic approach, a certain baseline has to be established if you want to build yourself up for a formal career in math, physics, or engineering. Let's build up a fundamental skill set that is going to build your fundamental math skills and help you master any subject. A few key areas I think students should focus on are outlined below.

The Importance of Algebra

If you study physics or engineering, algebra never goes away. So the first step on the road to becoming the next Stephen Hawking is to master this tedious yet fundamental subject. Do yourself a favor and pick up a decent algebra book and work through it. Do every problem so that by the end of the book, factoring equations, logarithms and other math basics are second nature for you. In the same way that lifting weights is going to make a football or basketball a better athlete when the games are actually played, mastering algebra will pay off later when you're doing your homework in dynamics or quantum theory.

Trigonometry

If you go on to become an electrical engineer and study circuit analysis or decide to master black hole physics, one fundamental area of business you'll have in common with your colleagues is trigonometry. Make sure you know your trig inside and out, learn what the trig functions really mean and master those pesky identities. Also don't over look this one crucial fact-trigonometry also provides a simple arena where you can learn how to prove and/or derive results. We all know that later, when you take advanced physics courses, you're going to see the words "show that" pop up frequently in your homework problems.

This is sure to cause headaches among the mere mortals amongst us, but it turns out you can improve your skills in this area in a non-threatening way by deriving trig identities. Instead of viewing the derivation of trig identities as a tedious obstacle, start to look at this as an opportunity. All trig books have homework problems where you have to derive an identity so pick up a trig book and do it until your blue in the face. Take it seriously and write up each proof as if you were submitting a short paper to a major journal. This will teach you how to go from point A to point B mathematically and how to write up a derivation in a formal way that will allow someone else to understand what's going on. If you do, later it will be easier to get through homework in advanced classes, you'll get better grades, and you'll develop a good foundation for writing up theoretical derivations for research papers.

Graphing Functions

While any function can be graphed easily on the computer or on a graphing calculator, it is very important to be able to graph a function on the fly with nothing more than a pencil and paper. The key abilities you want to focus on are developing an intuitive sense for how functions behave and learning how to focus on how functions behave in various limits. That is, how does a function look when the argument is small? How does it behave as the argument goes to infinity? Dig out your calculus book and review techniques that use the first and second derivative to graph a function. I review these extensively in my recent book "Calculus in Focus".

Series and Complex Numbers

In my opinion, understanding the series expansion of functions and the behavior of complex numbers can't be underestimated. If you want to understand physics, you need to master the use of series. Start by learning how to expand a function in a series. Some series should be second nature ('oh yeah, that's cosine"). Learn about convergence. Get a copy of Arfken and review the solution of differential equations using series. Try to get an intuitive feel for cutting a series off at a given term while retaining the essential behavior of the function. These are tools that are important when studying theoretical physics or advanced engineering.

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*Learning physics should be easy*

*Learning physics should be easy*

While its true that not all of us are Einsteins, should it be so difficult to learn math, science, and engineering that only a small handful of people can get degrees in these fields?Part of the problem is the way that math, physics, and engineering are taught. I haven't decided if there is a conscious conspiracy or not-but the truth is these subjects are generally taught in a way that is not helpful to most people. Maybe its because the vast majority of people that become professors are simply quite a bit smarter than the rest of us, and they don't realize what they're doing because they just "get it" and figure if you don't "get it" you are'nt cut out to be a physicist or mathematician.

In a typical college experience, I took "Electrodynamics" in graduate school. The professor was a great lecturer, but his lectures were really a complete waste of time. Basically, we spent our days in class listening to him spit out the book. He would recite the theorems and prove them. Had the book not been available his lectures would have been gold, but since we could buy a book, in fact since we were required to buy a book that had all this exact material in it, the lectures turned out to be no help at all. It's important to reinforce concepts to be sure, but physics, math and engineering are about doing things. These are active fields where problems must be solved. Its not about memorizing a theorem, its about applying it or being able to derive a new one.

Rather than "lecture", I would prefer that professors assign a book they are going to follow and then use the class time to help students solve problems. They should have the students read a given chapter before coming to class, and then spend class showing students how to solve some problems. Homework can then be assigned allowing students to build on what they did in class to learn how to solve the problems on their own.

Maybe physics professors like having a realm of mystery surround them. They like to feel smarter than everyone else and often aren't interested in helping people learn. So they keep problem solving tricks to themselves, and then tell the students who don't "get it" that they should become experimentalists or engineers. In Quantum Mechanics Demystified I have attempted to provide readers with format that makes learning physics straightforward. I show you how to solve quantum mechanics problems, and then you can try to do similar problems on your own.

From the fringes of quantum physics and relativity theory comes Bob Lazaar, an interesting man who claims to have worked on UFO's at Area 51. Tonight on coast to coast AM, the show will be guest hosted by George Knapp, a TV reporter who broke the Bob Lazaar story several years ago.

Lazaar is a very well spoken and charismatic man, but his story doesn't add up. I am not even talking about his wild claim to have worked on UFO's and "antigravity" propulsion at Area 51. Let's just start with his basic storyline.

*He claims to be a physicist that worked at several places where a clearance is required, including Los Alamos and Area 51. I don't recall where he claimed to have gone to school, it might have been Cal Tech or MIT, but something that stood out for me was he claimed not to recall any professors names.*

Anyone who has gotten a technical degree will recognize this claim as absurd. Students in math, physics and engineering run into hard professors and nutty professors, and good professors that just downright torture you during the semester. At least some of the names of these professors stick to you like glue throughout your life and its something that binds you to your fellow students at the institution where you got your degree. So when I heard Lazaar make this statement it struck me as odd to say the least.

Then there is the problem of his academic record. As I recall he claimed to have an advanced degree in physics yet there was no record of him having attended any of these prestigious instituions. This was explained by the claim that the CIA wiped out his record or something like that.

It has also been difficult to verify his work record. The sole evidence he worked at Los Alamos is a single paystub for a 2 week period where he worked as a technician. Again, I believe this is explained away by the vast powers in the CIA wiping out his record.

Anyone who believes the CIA or any government entity is that powerful or that places like Los Alamos are that secretive has been spending too much time watching television! The fact is its no secret who works at Los Alamos or any other government lab. The only things that are secret are the details of the project they work on. Its easy to find out that Joe Schmo works at X national lab in department Y. To me, the fact that no such record exists for Lazaar indicates that at best he worked as a temp here and there doing contract work. He was not some top scientist that would be called upon in the extremely unlikely event that they needed someone to "reverse engineer" a UFO. In a nutshell, I basically don't buy into Lazaar or his crazy story.

If anything, the Lazaar phenomenon is an interesting study in human behavior and our wish to live in a universe inhabited by aliens. These are all scientific issues and having a reasonable understanding of them is important. The population should have their own understanding of the issues to a certain degree rather than having to rely on experts for everything. This can only be done through the education system, and while these are conceptually based topics ultimately there is a mathematical underpinning. I am not suggesting that people should be going out and doing their own calculations of say uranium half-lives, but doing some calculations like that in school will allow someone to make more reasoned judgements on many issues-like storing nuclear wastes at Yucca Mountain.

Cohen says that algebra isn't a high or the highest form of human reasoning, and that writing is. Frankly Mr. Cohen I beg to differ. Mathematics is the highest form of human reasoning and is the basic underpinning of our modern society. It transcends the sciences, being at the root of the human genome project, the design of lasers, electric power, radio and cell phones and the internet. In short the entire modern world is fundamentally mathematically based. Writing was already highly developed long before calculus came into existence.

Cohen also claims that a computer or calculator can do math while they can't write. Cohen's understanding of how computers are used in mathematics is naive. In higher mathematics, doing a solution like the quadratic equation is trivial. Its the understanding of the solutions and properties of equations that require higher reasoning. The computer is used to churn out solutions to equations that are too hard for the human mind to solve directly (no analytic or closed form solution). In the end a human being has to analyze and interpret the results-something a computer can't do.

One of the themes touched on in "Equations of Eternity" by David Darling is the unreasonable effectiveness of mathematics in describing the physical world. Time and again, as Darling points out, mathematicians have worked on some obscure theoretical idea or area that seems to have nothing to do with reality. Then years later physicists stumble on it and discover that it describes some physical process in absolute detail, down to the last dotted i and crossed it.

A great example of the connection between mathematics and the physical world is the discovery by Maxwell of well, Maxwell's equations. During the 19th century the frontiers of science were being pushed by people studying electromagnetic phenomena. Years earlier Coloumb had figured out how to describe the electric force between two charges. In the early to mid-1800's physics had moved quite a bit beyond that to consider electric currents and magnetic fields. It was here that mathematical insight would prove to be an unusually effective tool-revealing properties of nature hidden to the senses.

In about the 1830's Ampere worked out a "law" that relates the magnetic field to a flow of current. Ampere's law has a very precise mathematical form which was worked out from careful experimental observation.

These "laws" of vector calculus are abstract mathematical laws--supposedly laws of pure thought. At first sight one might not expect that they would hold precedence over experimental observation. But it turns out they do. Maxwell used the laws to determine what form Ampere's law should really have, and in the process discovered something that was unknown at the time-radio waves.

This is just one small example of the interplay between math and physics. Later we'll explore connections between abstract mathematics and quantum theory which describes every last detail of atomic behavior.

The Holy Grail of physics is the unification of quantum physics and relativity, a Herculean task trying to wed together two spheres as different as night and day. On one hand we have the world of the very large. This is the world of stars, planets and galaxies-the world governed by Einstein’s relativity. On the other hand we have the world of the very small-the world of atoms, neutrons, and quarks-governed by the quantum. Each of these two realms not only describes different types of objects or different sized objects-they require different types of

*mathematics*. Even worse-the world of stars and galaxies seems to be governed by a classical, deterministic physics which fits neatly into a beautiful geometric theory, while the world of elementary particles is governed by probability, randomness, and mysterious mathematical worlds called Hilbert spaces-the world of the quantum dice.
At first glance these two theories can hardly be thought to be describing objects that belong to one and the same universe-but they do exactly that. Stars are made of atoms that obey the laws of quantum physics. Out of chance, chaos and ghostly entanglement-the orderly structure of a galaxy somehow emerges-and if we look at the ghostly quantum particles closely-atoms and elementary particles do fall in a gravitational field. Therefore there must be a path forward to a unified new physics.

Before embarking on the path of unification, it is important to make sure that one has a complete and thorough understanding of those two pillars of physics that are already well established-quantum physics and general relativity. This understanding is necessary before moving on to explore efforts at unification such as string theory and loop quantum gravity.

Let’s take a stab at quantum theory. In my book

*Quantum Mechanics Demystified*, I lay out the mathematical framework of quantum theory. But what is the conceptual framework-the basic building blocks that one wants to come away with before trying to put together a unified theory that describes the universe?.

__These are the ten keys to quantum physics.__They may not be the only ones-I am simply making suggestions of key concepts. You may wish to add your own. Before we start-a brief note on notation. We will denote the state of a particle or system with a bold capital letter, such as

**F**or

**G**, while a scalar (a plain old number) will be denoted by an small italic letter such as

*a*or

*b*.

**1. QUANTUM STATES CAN EXIST IN SUPERPOSITION**

One principle that plays a

*central,*absolutely vital role in quantum theory is the notion of*superposition.*Imagine if you will that a particle can be in two mutually exclusive states that we denote**F***and***G**. These two states could represent going through one slit or the other in the two slit experiment, or they could be two different energy states of an electron in an atom, for example. The principle of superposition tells us that the state formed by their linear combination-i.e. their sum-is also a valid quantum state. That is a system can be in the state described by**H**=*a***F**+*b***G**. A quantum superposition is a special sort of beast-when we look at the system, that is when we make a measurement, we never find it in some strange mixture of the states**F**and**G**. Rather, it is always in one state or the other. That is when the system is prepared in state**H**measurement will sometimes find the system in state**F**, while at other times, measurement will find the system in state**G**. Note that the numbers*a*and*b*can be complex. In key #2, we interpret the meaning of the state**H**and explore it further.

*2. THE BORN RULE*

The Born Rule tells us the probability of finding a quantum system in this state or that using a simple recipe. If a quantum system is in the state described by

**H**=*a***F**+*b***G**, then the probability that the system is found in state**F**when a measurement is made is found by squaring*a*while the probability that the system is found in state**G**when a measurement is made is found by squaring*b*. It is important not to confuse the fact that the Born rule tells us how to extract probabilities from a quantum state with the notion that the state is a mere statistical mixture. A superposition state like**H**leads to interference effects-like the fringes seen on the screen in the double slit experiment-something a statistical mixture can’t do.

*3.IDENTICAL PREPARATION OF STATES DOES NOT RESULT IN IDENTICAL MEAUREMENT RESULTS*

A key concept in classical science is that if you set up an experiment in exactly the same way several times- you will get repeatable results. The probabilistic nature of quantum theory-which is inherently fundamental and is not due to a lack of precision in our measurement devices-means that in many cases, if we prepare several systems in a given state, we will get different measurement results when the experiment is run. Once again, suppose that we prepare the system in the state

**H**=*a***F**+*b***G**. If we run the experiment 4 times, we might make measurements and find the results**F**,**F**,**G**,**F**. The next day if we again prepare systems in state**H**=*a***F**+*b***G**, we might instead get**F**,**G**,**G**,**G**. Now if we do the experiment a large number of times, then the relative fractions of**F**and**G**will tend to the probabilities given by squaring the coefficients*a*and*b*.

*4. THE UNCERTAINTY PRINCIPLE*

The Heisenberg uncertainty principle tells us that we cannot know the values of two complimentary observables with absolute precision. The quintessential example given in most textbooks is the uncertainty relation between position and momentum. In short, the uncertainty principle tells us that the more precision we use in measurement of position, the less we can know about momentum and vice versa. If we wish, we can know the value of one variable to any precision we like-but at the expense of complete uncertainty in the other variable. For example, if we choose to measure a particle’s momentum with great accuracy, then we sacrifice knowledge of the particles position. The uncertainty principle signs the death warrant of classical, deterministic physics.

*5. THE SCHRODINGER EQUATION AND UNITARY EVOLUTION*

Quantum states to evolve in time in a deterministic manner that is governed by the Schrodinger equation. Or in a more modern sense-the time evolution of quantum states is described by unitary evolution.

*6. LEARN LINEAR ALGEBRA*

The fifth key to quantum theory is that the world of quantum mechanics lies in the mathematical realm of vector spaces. This means that the mathematics of quantum physics is the mathematics of linear algebra. If you want to master quantum theory then you need to know linear algebra. Learn how to manipulate matrices, how to calculate determinants, and how to find eigenvectors and eigenvalues. Learn about abstract vector spaces. Finally, learn about special types of matrices, in particular Hermitian and Unitary matrices.

*6. PHYSICAL OBSERVABLES ARE REPRESENTED BY OPERATORS*

This follows on the heels of point 6. In quantum theory, physical observables like momentum and energy are represented by operators. When considering the wave function approach of the Schrodinger equation, operators are instructions to do something to a function-compute the derivative say. In the mathematically equivalent matrix mechanics derived by Heisenberg, operators are represented by matrices.

*7. THE ONLY POSSIBLE RESULTS OF A MEASUREMENT ARE THE EIGENVALUES OF AN OPERATOR*

Following key #7, since physical observables are represented by mathematical operators, the next logical question to ask is what are the possible measurement results as predicted by the theory? It turns out that these are the eigenvalues of the operators used to represent physical observables.

*8. WHAT IS THE MEANING OF THE WAVEFUNCTION?*

An important issue in quantum theory is the following. Is the wavefunction a real, physical entity? Or does it just represent our state of knowledge? The meaning of the wavefunction is an important issue to resolve before we can have a “theory of everything”.

*9. DECOHERENCE*

Quantum systems interact with their environment. In doing so, the strange quantumness of their nature is lost. That is superposition and the interference that comes along with it is washed out by interactions with the environment. It is believed by many that this is how the classical world of our senses arises out of the quantum morass.

*10. ENTANGLEMENT*

Quantum physics is filled with mysteries and perhaps the greatest mystery of all is that of

*entanglement*-the spooky action at a distance type correlation originally put forward by Einstein and his colleagues back in 1935. If two particles A and B are entangled, then their properties become correlated. In a certain state, if you measure the spin of A and find it to be spin-up, then the spin of B is spin-down. Or if you measure A and find it to be spin-down, then B is spin-up. If that is too abstract to wrap your mind around, for a loose analogy imagine that A and B are entangled dice correlated such that they always roll the same number. Roll A and find a 3, then rolling B is guaranteed to give a 3.
Spooky action at a distance tells us that we can leave A here on earth and carry B all the way to the other side of the galaxy-and their measurement results will remain correlated. Of course this isn’t completely worked out, even assuming the particles could be protected from the environment, relativistic effects and gravity might hamper such a scenario, but spatial separation, i.e. distance alone doesn’t seem to have an effect on entanglement.

The type of correlation that results in entanglement is a bit spooky in itself. It’s not really mere correlation. Without diving into the mathematics, suffice it to say that basically the two particles loose the essence of their individual identities. In an entangled system, the system as a whole assumes an identity.

It is as if the whole becomes greater than the sum of its parts. To understand the entangled system you need to understand the whole and cannot understand it from the individual components alone. This amazing phenomenon-which has been confirmed in the laboratory-is of central importance in quantum computation and quantum cryptography. Quantum physics can be mathematically daunting. Before you go on to tackle string theory and quantum gravity-make sure you have these central ideas down